polykin.transport.hmt

Nu_plate_free

Nu_plate_free(
    orientation: Literal[
        "vertical",
        "horizontal-upper-heated",
        "horizontal-lower-heated",
    ],
    Ra: float,
    Pr: float | None = None,
) -> float

Calculate the Nusselt number for free convection on a vertical or horizontal plate.

For a vertical plate of height \(L\), the average Nusselt number \(\overline{Nu}=\bar{h}L/k\) is estimated by the following expression:

\[ \overline{Nu} = \left(0.825 + \frac{0.387 Ra^{1/6}} {[1 + (0.492/Pr)^{9/16}]^{8/27}}\right)^2 \]

where \(Ra\) is the Rayleigh number based on the plate height and \(Pr\) is the Prandtl number.

If the plate is horizontal, the flow and heat transfer patterns depend on whether the surface is heated or cooled, and which direction it is facing. For the upper surface of a heated plate (or the lower surface of a cooled plate), the Nusselt number is estimated by the following expression:

\[ \overline{Nu} = \begin{cases} 0.54 Ra^{1/4} ,& 10^4 \lesssim Ra \lesssim 10^7 \\ 0.15 Ra^{1/3} ,& 10^7 \lesssim Ra \lesssim 10^{11} \end{cases} \]

For the lower surface of a heated plate (or the upper surface of a cooled plate), the Nusselt number is estimated by the following expression:

\[ \overline{Nu} = 0.27 Ra^{1/4} \;, 10^5 \lesssim Ra \lesssim 10^{10} \]

where \(Ra\) is the Rayleigh number based on the ratio between the plate surface area and perimeter.

In all cases, the properties are to be evaluated at the film temperature.

Tip

• The correlation for vertical plates can also be applied to vertical cylinders of height \(L\) and diameter \(D\) if \(D/L \gtrsim 35/Gr_L^{1/4}\).
• The correlation for vertical plates can also be applied to the top and bottom surfaces of heated and cooled inclined plates, respectively, if \(g\) is replaced by \(g \cos \theta\) in the calculaton of \(Ra\).

References

• Incropera, Frank P., and David P. De Witt. "Fundamentals of heat and mass transfer", 4th edition, 1996, p. 493-498.

PARAMETER	DESCRIPTION
orientation	Orientation of the plate. TYPE: Literal['vertical', 'horizontal-upper-heated', 'horizontal-lower-heated']
Ra	Rayleigh number based on plate characteristic length. For a vertical plate, the characteristic length is the plate height. For a horizontal plate, the characteristic length is the ratio between the plate surface area and perimeter. TYPE: float
Pr	Prandtl number. TYPE: float DEFAULT: None

RETURNS	DESCRIPTION
float	Nusselt number.

See also

• Nu_plate: related method for forced convection.

Examples:

Estimate the heat transfer coefficient between the outer surface of a vertical tank (D=2 m, H=3 m) with a surface temperature of 310 K and quiescent air at 290 K.

>>> from polykin.transport import Nu_plate_free
>>> L = 3.0           # m (characteristic length is tank height)
>>> Ts, Tb = 310, 290 # K
>>> Tf = (Ts + Tb)/2  # K (film temperature)
>>> Pr = 0.707        # m (properties at Tf=300 K)
>>> nu = 15.9e-6      # m²/s
>>> k = 26.3e-3       # W/m/K
>>> beta = 1/Tf       # 1/K
>>> g = 9.81          # m/s²
>>> Gr = g*beta*(Ts - Tb)*L**3/nu**2
>>> Ra = Gr*Pr
>>> Nu = Nu_plate_free('vertical', Ra, Pr)
>>> h = Nu*k/L
>>> print(f"h={h:.1e} W/m².K")
h=3.7e+00 W/m².K

Source code in src/polykin/transport/hmt.py

def Nu_plate_free(orientation: Literal['vertical',
                                       'horizontal-upper-heated',
                                       'horizontal-lower-heated'],
                  Ra: float,
                  Pr: float | None = None
                  ) -> float:
    r"""Calculate the Nusselt number for free convection on a vertical or
    horizontal plate.

    For a vertical plate of height $L$, the average Nusselt number 
    $\overline{Nu}=\bar{h}L/k$ is estimated by the following expression:

    $$  \overline{Nu} = \left(0.825 + \frac{0.387 Ra^{1/6}}
        {[1 + (0.492/Pr)^{9/16}]^{8/27}}\right)^2 $$

    where $Ra$ is the Rayleigh number based on the plate height and $Pr$ is the
    Prandtl number.

    If the plate is horizontal, the flow and heat transfer patterns depend on
    whether the surface is heated or cooled, and which direction it is facing.
    For the upper surface of a heated plate (or the lower surface of a cooled
    plate), the Nusselt number is estimated by the following expression:

    $$ \overline{Nu} =
    \begin{cases}
    0.54 Ra^{1/4} ,& 10^4 \lesssim Ra \lesssim 10^7 \\
    0.15 Ra^{1/3} ,& 10^7 \lesssim Ra \lesssim 10^{11}
    \end{cases} $$

    For the lower surface of a heated plate (or the upper surface of a cooled
    plate), the Nusselt number is estimated by the following expression:

    $$ \overline{Nu} = 0.27 Ra^{1/4} \;, 10^5 \lesssim Ra \lesssim 10^{10} $$

    where $Ra$ is the Rayleigh number based on the ratio between the plate
    surface area and perimeter.

    In all cases, the properties are to be evaluated at the film temperature.

    !!! tip

        * The correlation for vertical plates can also be applied to vertical 
        cylinders of height $L$ and diameter $D$ if $D/L \gtrsim 35/Gr_L^{1/4}$.
        * The correlation for vertical plates can also be applied to the top and
        bottom surfaces of heated and cooled _inclined_ plates, respectively, 
        if $g$ is replaced by $g \cos \theta$ in the calculaton of $Ra$. 

    **References**

    * Incropera, Frank P., and David P. De Witt. "Fundamentals of heat and
      mass transfer", 4th edition, 1996, p. 493-498.

    Parameters
    ----------
    orientation: Literal['vertical', 'horizontal-upper-heated', 'horizontal-lower-heated']
        Orientation of the plate.
    Ra : float
        Rayleigh number based on plate characteristic length. For a vertical
        plate, the characteristic length is the plate height. For a horizontal
        plate, the characteristic length is the ratio between the plate surface
        area and perimeter.
    Pr : float
        Prandtl number.

    Returns
    -------
    float
        Nusselt number.

    See also
    --------
    - [`Nu_plate`](Nu_plate.md): related method for forced convection.

    Examples
    --------
    Estimate the heat transfer coefficient between the outer surface of a
    vertical tank (D=2 m, H=3 m) with a surface temperature of 310 K and
    quiescent air at 290 K.
    >>> from polykin.transport import Nu_plate_free
    >>> L = 3.0           # m (characteristic length is tank height)
    >>> Ts, Tb = 310, 290 # K
    >>> Tf = (Ts + Tb)/2  # K (film temperature)
    >>> Pr = 0.707        # m (properties at Tf=300 K)
    >>> nu = 15.9e-6      # m²/s
    >>> k = 26.3e-3       # W/m/K
    >>> beta = 1/Tf       # 1/K
    >>> g = 9.81          # m/s²
    >>> Gr = g*beta*(Ts - Tb)*L**3/nu**2
    >>> Ra = Gr*Pr
    >>> Nu = Nu_plate_free('vertical', Ra, Pr)
    >>> h = Nu*k/L
    >>> print(f"h={h:.1e} W/m².K")
    h=3.7e+00 W/m².K
    """
    if orientation == 'vertical':
        if Pr:
            return (0.825 + 0.387*Ra**(1/6) / (1 + (0.492/Pr)**(9/16))**(8/27))**2
        else:
            raise ValueError("`Pr` is required for vertical plates.")
    elif orientation == 'horizontal-upper-heated':
        check_range_warn(Ra, 1e4, 1e11, 'Ra')
        if Ra < 1e7:
            return 0.54*Ra**(1/4)
        else:
            return 0.15*Ra**(1/3)
    elif orientation == 'horizontal-lower-heated':
        check_range_warn(Ra, 1e5, 1e10, 'Ra')
        return 0.27*Ra**(1/4)
    else:
        raise ValueError(f"Unknown `orientation`: {orientation}.")